The 85 Foldings of the Latin Cross

(Erik Demaine, Martin Demaine, Anna Lubiw, Joseph O'Rourke)

As part of our work
on folding polygons into convex polyhedra, we are interested in exploring
the many possibilities for seemingly simple examples. Here we look at the
Latin cross, which is made up of 6 unit squares. It is well-known that this
cross folds into a cube. What is amazing is that it folds into many other
shapes.

It is even more surprising how much more is possible with non-edge-to-edge
gluings. Koichi Hirata
wrote an excellent
program
to compute all the possible gluings of a given polygon into convex
polyhedra. The results are shown graphically below.
This list was independently verified by a program written by Anna Lubiw.
We have tried all of these gluings by hand, and determined the
(unique) crease patterns that permit folding into convex polyhedra.
These are also shown below.

The following gluings and foldings are in no particular order.
In each case, the gluing is shown (equal numbers are glued together),
and the crease pattern is shown if it is "unique."
Crease patterns can be clicked on to obtain printable PostScript.

All gluings except the cube come in symmetric pairs, because the Latin
cross has reflectional symmetry. Thus, some gluings are simply
marked ``Gluing symmetric to n'' This means that the gluing
and the crease pattern can be obtained simply by reflecting those for n.
In each case, we also give a brief description of the resulting polytope.
When it is the same as a previous polytope, or the mirror reflection of
a previous polytope, this is noted. (This characterization is all done by
hand, but we have double-checked that it is correct.)
The result is 21 distinct polytopes that can be folded from the Latin cross.

1. Cube [edge-to-edge]

2. Octahedron, six degree-4 vertices

3. Reflectionally symmetric pentahedron, six degree-3 vertices

4. Same polytope as 3

5. Octahedron [edge-to-edge]

6. Mirror polytope of 2

7. Same polytope as 3

8. Same polytope as 3

9. Mirror polytope of 2

10. Same polytope as 2

11. Hexahedron, three degree-4 vertices and two degree-3 vertices

12. Reflectionally symmetric pentahedron [edge-to-edge]

13. Same polytope as 12 [edge-to-edge]

14. Mirror polytope of 11

15. Mirror polytope of 5 [edge-to-edge]

16. Gluing symmetric to 9

17. Gluing symmetric to 3

18. Hexahedron, three degree-4 vertices and two degree-3 vertices

19. Tetrahedron

20. Pyramid with quadrangular base

21. Mirror polytope of 2

22. Same polytope as 20

23. Gluing symmetric to 7

24. Hexahedron, three degree-4 vertices and two degree-3 vertices

25. Mirror polytope of 20

26. Gluing symmetric to 18

27. Gluing symmetric to 15 [edge-to-edge]

28. Same polytope as 11

29. Same polytope as 34 [edge-to-edge]

30. Doubly covered quadrangle [edge-to-edge]

31. Octahedron, two degree-5 vertices, two degree-4 vertices, and two degree-3 vertices

32. Octahedron, two degree-5 vertices, two degree-4 vertices, and two degree-3 vertices

33. Mirror polytope of 32

34. Tetrahedron [edge-to-edge]

35. Gluing symmetric to 29 [edge-to-edge]

36. Gluing symmetric to 10

37. Gluing symmetric to 4

38. Gluing symmetric to 21

39. Same polytope as 3

40. Gluing symmetric to 8

41. Gluing symmetric to 2

42. Gluing symmetric to 25

43. Reflectionally symmetric tetrahedron

44. Same polytope as 24

45. Gluing symmetric to 39

46. Same polytope as 18

47. Gluing symmetric to 6

48. Same polytope as 20

49. Gluing symmetric to 44

50. Gluing symmetric to 20

51. Gluing symmetric to 14

52. Tetrahedron ("almost tetrapack")

53. Tetrahedron

54. Mirror polytope of 19

55. Gluing symmetric to 52

56. Gluing symmetric to 28

57. Gluing symmetric to 13 [edge-to-edge]

58. Gluing symmetric to 54

59. Gluing symmetric to 34 [edge-to-edge]

60. Same polytope as 30 [edge-to-edge]

61. Doubly covered quadrangle

62. Gluing symmetric to 43

63. Gluing symmetric to 19

64. Octahedron, two degree-5 vertices, two degree-4 vertices,
and two degree-2 vertices